RHS criterion for right triangles

Right triangles have a special criterion that does not work for general triangles: knowing the hypotenuse and one leg is enough.

Idea

RHS (Right angle–Hypotenuse–Side) criterion. If, in two right triangles, the hypotenuses are equal and any one pair of legs is equal, then the triangles are congruent.

In symbols: if $\angle B = \angle E = 90^\circ$, hypotenuse $AC =$ hypotenuse $DF$, and one leg $AB = DE$, then $\triangle ABC \cong \triangle DEF$.

Why does it work? Once the right angle, hypotenuse and one leg are fixed, the other leg is automatically determined (by Pythagoras: $\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2$). So all three sides are essentially fixed , which is just SSS. RHS is a shortcut for right triangles.

Why isn't RHS just SSA? Notice that SSA in general fails because the swing of the third side can hit the base in two places. For a right triangle, the right angle constraint means one of those swing positions becomes impossible , leaving only one. So SSA, which fails generally, succeeds for right triangles. That special case is what we call RHS.

This criterion comes up often in geometry: many proofs hinge on showing two right triangles have equal hypotenuse and one equal leg.

Worked examples

Example 1. In two right triangles, hypotenuses are $10$ cm each and one leg is $6$ cm each. Congruent?

By RHS, yes. (The other leg, by Pythagoras, must be $\sqrt{10^2 - 6^2} = \sqrt{64} = 8$ in both.)

Example 2. Right triangles with hypotenuse $13$ and one leg $5$ , congruent? What is the other leg?

By RHS, yes. The other leg: $\sqrt{13^2 - 5^2} = \sqrt{144} = 12$.

Example 3. Two right triangles share a non-hypotenuse leg of $4$, and the second leg is $3$ in one and $5$ in the other. Congruent?

No. The right angle is included between the two legs; so two legs $+$ right angle is SAS , but the legs differ, so triangles differ.

Example 4. In a kite (two congruent triangles sharing a base), if the central diagonal is drawn perpendicular to the base, prove the two right triangles formed are congruent by RHS.

They share the perpendicular leg (the diagonal), have equal hypotenuses (the two equal sides of the kite), and both have a right angle. So RHS applies.

Try it yourself

1. State the RHS criterion in symbols.
2. Two right triangles with hypotenuse $5$ and one leg $3$. Congruent?
3. Right triangle hypotenuse $25$, one leg $7$ , find the other leg.
4. Two right triangles share an acute angle and a leg. Congruent? Which criterion?
5. Why does RHS not apply to non-right triangles?
6. State a real-life example where RHS is useful.
7. A right triangle has legs $5, 12$. Hypotenuse?
8. RHS is a special case of which general criterion?

Activity

Cut out two right triangles with the same hypotenuse and one matching leg. Place them on top of each other; verify they coincide.